Abstract
Very recently, Samet et al. (Int. J. Anal. 2013:917158, 2013) and Jleli-Samet (Fixed Point Theory Appl. 2012:210, 2012) noticed that some fixed point theorems in the context of a G-metric space can be deduced by some well-known results in the literature in the setting of a usual (quasi) metric space. In this paper, we note that the approach of Samet et al. (Int. J. Anal. 2013:917158, 2013) and Jleli-Samet (Fixed Point Theory Appl. 2012:210, 2012) is inapplicable unless the contraction condition in the statement of the theorem can be reduced into two variables. For this purpose, we modify some existing results to suggest new fixed point theorems that fit with the nature of a G-metric space. The expressions in our result, the contraction condition, cannot be expressed in two variables, therefore the techniques used in (Int. J. Anal. 2013:917158, 2013; Fixed Point Theory Appl. 2012:210, 2012) are not applicable.
MSC:47H10, 54H25.
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1 Introduction
The concept of G-metric space was introduced by Mustafa and Sims [1] in order to extend and generalize the notion of metric space. In this paper, the authors characterized the Banach contraction mapping principle [2] in the context of a G-metric space. Following this initial report, a number of authors have characterized many well-known fixed point theorems in the setting of G-metric space (see, e.g., [1, 3–36]). Since one is adapted from the other, there is a close relation between a usual metric space and a G-metric space (see, e.g., [1, 23–27]). In fact, the nature of a G-metric space is to understand the geometry of three points instead of two points via perimeter of a triangle. However, most of the published papers dealing with a G-metric space did not give much importance to these details. Consequently, a great majority of results were obtained by transforming the contraction conditions from the usual metric space context to a G-metric space without carrying enough of the characteristics of the G-metric.
Very recently, Samet et al. [37] and Jleli-Samet [38] observed that some fixed point theorems in the context of a G-metric space in the literature can be concluded by some existing results in the setting of a (quasi-)metric space. In fact, if the contraction condition of the fixed point theorem on a G-metric space can be reduced to two variables instead of three variables, then one can construct an equivalent fixed point theorem in the setup of a usual metric space. More precisely, in [37, 38], the authors noticed that forms a quasi-metric. Hence, if one can transform the contraction condition of existence results in a G-metric space in such terms, , then the related fixed point results become the known fixed point results in the context of a quasi-metric space.
In this paper, we notice that the techniques used in [37, 38] are valid if the contraction condition in the statement of the theorem can be expressed in two variables. Furthermore, we prove some fixed point theorems in the context of a G-metric space for which the techniques in [37, 38] are inapplicable.
2 Preliminaries
In this section we recollect basic definitions and a detailed overview of the fundamental results. Throughout this paper, ℕ is the set of nonnegative integers, and is the set of positive integers.
Definition 2.1 (See [1])
Let X be a non-empty set and let be a function satisfying the following properties:
-
(G1)
if ,
-
(G2)
for all with ,
-
(G3)
for all with ,
-
(G4)
(symmetry in all three variables),
-
(G5)
for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Every G-metric on X defines a metric on X by
Example 2.1 Let be a metric space. The function , defined as
or
for all , is a G-metric on X.
Definition 2.2 (See [1])
Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if
that is, for any , there exists such that for all . We call x the limit of the sequence and write or .
Proposition 2.1 (See [1])
Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x,
-
(2)
as ,
-
(3)
as ,
-
(4)
as .
Definition 2.3 (See [1])
Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there is such that for all , that is, as .
Proposition 2.2 (See [1])
Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy,
-
(2)
for any , there exists such that for all .
Definition 2.4 (See [1])
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
We will use the following result which can be easily derived from the definition of a G-metric space (see, e.g., [1]).
Lemma 2.1 Let be a G-metric space. Then
Definition 2.5 (See [1])
Let be a G-metric space. A mapping is said to be G-continuous if is G-convergent to where is any G-convergent sequence converging to x.
In [22], Mustafa characterized the well-known Banach contraction mapping principle in the context of G-metric spaces in the following ways.
Theorem 2.1 (See [22])
Let be a complete G-metric space and let be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 2.2 (See [22])
Let be a complete G-metric space and let be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Remark 2.1 The condition (2) implies the condition (3). The converse is true only if . For details, see [22].
Theorem 2.3 (See [26])
Let be a G-metric space. Let be a mapping such that
for all x, y, z, where a, b, c, d are positive constants such that . Then there is a unique such that .
Theorem 2.4 (See [27])
Let be a G-metric space. Let be a mapping such that
for all x, y, z, where . Then there is a unique such that .
Theorem 2.5 (See [26])
Let be a G-metric space. Let be a mapping such that
for all x, y, z, where a, b are positive constants such that . Then there is a unique such that .
Theorem 2.6 (See [26])
Let be a G-metric space. Let be a mapping such that
for all x, y, z, where a, b are positive constants such that . Then there is a unique such that .
Theorem 2.7 (See [25])
Let be a G-metric space. Let be a mapping such that
for all x, y, z, where . Then there is a unique such that .
Theorem 2.8 (See, e.g., [38])
Let be a complete G-metric space and let be a given mapping satisfying
for all , where is continuous with . Then there is a unique such that .
Definition 2.6 (See, e.g., [38])
A quasi-metric on a nonempty set X is a mapping such that
(p1) if and only if ,
(p2) ,
for all . A pair is said to be a quasi-metric space.
Samet et al. [37] and Jleli-Samet [38] noticed that is a quasi-metric whenever is a G-metric. It is well known that each quasi-metric induces a metric. Indeed, if is a quasi-metric space, then the function defined by
is a metric on X.
Theorem 2.9 Let be a complete metric space and let be a mapping with the property
for all , where q is a constant such that . Then T has a unique fixed point.
Samet et al. [37] proved that Theorem 2.4-Theorem 2.7 are the consequences of Theorem 2.9 by using the following proposition.
Proposition 2.3
-
(A)
If is a complete G-metric space, then is a complete metric space.
-
(B)
If is a sequentially G-compact G-metric space, then is a compact metric space.
3 Main results
We first state the following theorem about the existence and uniqueness of a common fixed point, which is a generalization of Theorem 2.7. Furthermore, the techniques of the papers [37, 38] are not applicable to this theorem.
Theorem 3.1 Let be a G-metric space. Let be a mapping such that
for all x, y, z, where and
Then there is a unique such that .
Proof Let . We define a sequence in the following way:
Taking , in (11), we find
where
Now, we have to examine four cases in (14). For the first case, assume that . Then the expression (13) turns into
It is a contradiction since . For the second case, assume that . Regarding (G5) together with the inequality (13), we derive that
a contradiction since .
For the third case, assume that . By (G5) and the inequality (13), we have
which is equivalent to
where since .
For the last case, assume that . Then the inequality (13) turns into
where .
As a result, from (15)-(19) we conclude that
where and hence . We show that the sequence is G-Cauchy. By the rectangle inequality (G5), we have for
Letting in (21), we get that . Hence, is a G-Cauchy sequence in X. Since is G-complete, then there exists such that is G-convergent to . We shall show that . Suppose, on the contrary, that . On the other hand, we have and hence
where
Letting in (22) and using the fact that the metric G is continuous, we get that either
or
by the rectangular property (G5). Since , the inequalities above yield contradictions. Hence we have , that is, .
Finally, we shall show that is the unique fixed point of T. Suppose that contrary to our claim, there exists another common fixed point with . From (4) we have
where
Hence, the inequality (25) is equal to either
or
Since , the expressions (26) and (27) yield contradictions. Thus, is the unique fixed point of T. □
In Theorem 3.1, the interval of constant of the contractive condition can be extended to the interval by eliminating the same terms. Since the proof is the mimic of Theorem 3.1, we omit it.
Theorem 3.2 Let be a G-metric space. Let be a mapping such that
for all x, y, z, where and
Then there is a unique such that .
Remark 3.1 Theorem 2.1-Theorem 2.6 are the consequences of Theorem 3.1 and Theorem 3.2.
Inspired by Theorem 2.8, we state the following theorem for which the methods in [37, 38] are not applicable.
Theorem 3.3 Let be a complete G-metric space and let be a given mapping satisfying
for all , where is continuous with . Then there is a unique such that .
Proof We first show that if the fixed point of the operator T exists, then it is unique. Suppose, on the contrary, that z, w are two fixed points of T such that . Hence, . By (29), we have
which is equivalent to
a contradiction. Hence, T has a unique fixed point.
Let . We define a sequence in the following way:
If for some , then we get the desired result. From now on, we assume that for some . Taking , in (29), we find
Hence, is a positive decreasing sequence. Thus, the sequence converges to . We shall show that . Suppose, on the contrary, that . Letting in (33), we find that
It is a contradiction. Hence, we conclude that
Moreover, by Lemma 2.1, we derive that
Now, we demonstrate that the sequence is G-Cauchy. Suppose that is not G-Cauchy. So, there exists and subsequences and of with such that
Furthermore, corresponding to , one can choose such that it is the smallest integer with satisfying (37). Thus, we have
By the triangle inequality, we get
Letting in the expression (39) and keeping (36) in mind, we find
On the other hand, we have
and
Letting in the expression (41)-(42) and regarding (35), (36) and (40), we derive
Further, we have
by (G3) and the triangle inequality. Letting in (44) and using (35), (36) and (40), we conclude that
Analogously, we have
by (G3) and the triangle inequality. Letting in (44) and using (35), (36) and (43), we conclude that
Due to (33) and regarding (G4), we obtain
for all . Letting in the inequality (48) and keeping (45) and (47) in mind, we get
a contradiction. Hence, is a G-Cauchy sequence. Since is G-complete, there is such that .
We claim that . From (33), we have
Letting in (50), regarding the continuity of G, we get that
Hence , that is, . □
Remark 3.2 Let X be a nonempty set. We define functions in the following way:
for all . It is easy to see that both mappings p and q do not satisfy the conditions of Definition 2.6. Hence, Theorem 3.1 and Theorem 3.3 cannot be characterized in the context of quasi-metric as it is suggested in [37, 38].
Example 3.1 Let , be defined by
Then is a G-complete G-metric space. Let be defined by
and for all .
Proof For the proof the Example 3.1, we examine the following cases:
-
Let . Then
-
Let . Then
-
Let . Then
-
Let . Then
Then
Then the conditions of Theorem 3.3 hold and T has a unique fixed point. Notice that is the desired fixed point of T. □
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Acknowledgements
The authors express their gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions. The first author thank to student Peyman Salimi for his help for Example 3.1.
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Karapınar, E., Agarwal, R.P. Further fixed point results on G-metric spaces. Fixed Point Theory Appl 2013, 154 (2013). https://doi.org/10.1186/1687-1812-2013-154
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DOI: https://doi.org/10.1186/1687-1812-2013-154